§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:59:22Z

Audit 4.15 · Audit: Theorem T, twisted HH of \(\mathcal C_k\) is \(C_K\), and retract classes

proved

Theorem 1.3 is correct, but three steps lean on the wrong reference or on an unproved twisted variant. Equivariant formality of \(C^*(BL_\mu)\) does not follow from Chen–Yi's Theorem 93; it does follow from free even cohomology with scalar Frobenius, and then the Koszul computation (degree \(0\), one-dimensional) is right. Additivity of twisted HH along the recollement is asserted with 'the same proof'. LT must be defined by functoriality, not by a degree-\(0\) check. The Barr–Beck, Morita and cyclicity steps check out.

Audit 4.15.

Under (R), (C), D0–D3: (a) \(\mathrm{HH}(\mathcal C_\mu,\mathrm{Fr})=\overline{\mathbb Q}_\ell[\underline{\overline{\mathbb Q}_\ell},\mathrm{can}]\) in degree \(0\) for each orbit; \(\mathrm{HH}(\mathcal C_k,\mathrm{Fr})=\bigoplus_\mu\overline{\mathbb Q}_\ell[\Delta_\mu,\mathrm{can}]\) in degree \(0\); \(\mathrm{LT}\) is an isomorphism onto \(C_K\otimes\overline{\mathbb Q}_\ell\); likewise \(\mathrm{HH}(\mathcal H,\mathrm{Fr})\cong\mathcal H_K\otimes\overline{\mathbb Q}_\ell\). (b) \(\mathrm{HH}_0(\mathrm{act})\) is \(-\ast\varphi_0\); so (SG) \(\iff\mathrm{HH}_0(\mathrm{act})\) onto. (c) \([M,e\Phi]=[N,\pi\Phi\,\mathrm{Fr}^*\iota]\) for a Frobenius-equivariant idempotent \(e=\iota\pi\); the \([\mathrm{IC}_\nu,\mathrm{can}]\) form a basis (no hypothesis needed for this), and under \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) each is the class of a retract of a Satake object. (d) \(\operatorname{End}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{Hom}(V_\lambda,V_\lambda\ast A)\), and \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\iff\operatorname{Ind}\mathcal C_k\simeq\) modules over \(A\) in \(\operatorname{Ind}\mathcal H\).

Outcome: holds with minor defects (citation and rigor gaps, all repairable). (1) Formality: the cited Chen–Yi Thm 93 concerns the algebra \(A_X\) of their (41), not \(C^*(BL_\mu)\); repair in step 2. (2) The twisted additivity along \(\langle i_*\mathcal C_{Z\smallsetminus\mathcal O},j_!\mathcal C_\mu\rangle\) is asserted, not proved, and the cited Kuznetsov/Keller papers treat untwisted HH; it is a standard trace-additivity statement, sketched in step 3. (3) Well-definedness of LT on all of HH needs functoriality (step 4). (4) The literature on categorical traces in step 7 (AGKRRV, Gaitsgory–Varshavsky, Zhu, Ben-Zvi–Francis–Nadler, Ben-Zvi–Nadler) is cited without result numbers and is not local, so it cannot be checked here; the agent states it is unused, and nothing in (a)–(d) depends on it.

  1. One orbit: the category. \(\mathcal C_\mu\simeq D^b_{\operatorname{Stab}}(\mathrm{pt})\simeq D^b_{L_\mu}(\mathrm{pt})\) (pro-unipotent part contractible), \(L_\mu=Z_{G^\theta}(\mu)\) connected by (C). Every object has constant cohomology sheaves, so it is an iterated extension of shifts of \(\underline{\overline{\mathbb Q}_\ell}\), and \(\mathcal C_\mu=\operatorname{Thick}(\underline{\overline{\mathbb Q}_\ell})\simeq\operatorname{Perf}C^*(BL_\mu)\) (HA Thm 7.1.2.1, Schwede–Shipley; checked in the local copy), compatibly with \(\mathrm{Fr}^*\) via \(\mathrm{can}\).
  2. One orbit: formality and Koszul (repair of the citation). \(H^*(BL_\mu)=\operatorname{Sym}V\) is a polynomial algebra on even generators. Under (R), \(L_\mu\supseteq T^\theta\) is split (Audit 4.18 step 2), so Frobenius acts on \(H^{2d}\) by the scalar \(q^{d}\) (or \(q^{-d}\), depending on convention). Choose cocycle representatives \(s:V\to C^*(BL_\mu)\). Over \(\overline{\mathbb Q}_\ell\) the induced map of commutative algebras \(\operatorname{Sym}V\to C^*(BL_\mu)\) is a quasi-isomorphism. \(\mathrm{Fr}\circ s\) and \(s\circ\varphi_V\) induce the same map on cohomology, and \(V\) has zero differential, so they are homotopic. For the monoid \(\mathbb N\) generated by \(\mathrm{Fr}\), one homotopy is all the coherence needed, so this is an equivalence of algebras-with-endomorphism. Twisted HH is invariant under such equivalences and under Morita equivalence. With the Koszul resolution of \(\operatorname{Sym}V\) over \(\operatorname{Sym}V^{\otimes2}\), \(\mathrm{HH}(\operatorname{Sym}V,(\operatorname{Sym}V)_\varphi)\) is the Koszul complex on \((1-q^{\pm d_j})v_j\). This is a regular sequence because \(q^{d_j}\ne1\), so HH is \(\operatorname{Sym}V/(V)=\overline{\mathbb Q}_\ell\), concentrated in degree \(0\). The Koszul complex is finite, so totalization causes no convergence issue.
  3. Gluing (the unproved twisted step, sketched). \(\operatorname{Hom}(j_!-,i_*-)=\operatorname{Hom}(-,j^!i_*-)=0\) gives the semi-orthogonal decomposition, and \(\mathrm{Fr}^*\) preserves both pieces because orbits are Frobenius-stable. The diagonal bimodule of \(\mathcal C_Z\) sits in a triangle whose outer terms are the diagonal bimodules of the two pieces (through the projection functors). Tensoring with the bimodule \(\operatorname{Hom}(\mathrm{Fr}^*-,-)\), which is compatible with the decomposition, gives a split triangle \(\mathrm{HH}(i_*\mathcal C_{Z\smallsetminus\mathcal O},\mathrm{Fr})\to\mathrm{HH}(\mathcal C_Z,\mathrm{Fr})\to\mathrm{HH}(\mathcal C_\mu,\mathrm{Fr})\). This is the additivity of traces for an endofunctor preserving a semi-orthogonal decomposition; it is standard but is neither proved in Theorem 1.3 nor literally the cited results. HH commutes with the filtered union of the full subcategories \(\mathcal C_Z\). Since \(\mathcal O_\mu(\mathbb F_q)\) is one \(K\)-orbit, \(\mathrm{LT}[\Delta_\mu,\mathrm{can}]=\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}\), a basis: LT is an isomorphism. The Schubert-cell version for \(\mathcal H\) is identical (stabilizers \(Z_G(\lambda)\ltimes\)pro-unipotent, split).
  4. LT is well defined (repair). Checking the cyclicity relation on degree-\(0\) generators does not define a map out of the Hochschild complex. Instead, for \(x\in X(F)\) the stalk functor \(i_x^*:\mathcal C_k\to\operatorname{Perf}\overline{\mathbb Q}_\ell\) commutes with \(\mathrm{Fr}^*\) (as \(x\) is rational), so functoriality gives \(\mathrm{HH}(\mathcal C_k,\mathrm{Fr})\to\mathrm{HH}(\operatorname{Perf}\overline{\mathbb Q}_\ell,\mathrm{id})=\overline{\mathbb Q}_\ell\). On \([M,\beta]\) it is the graded trace of \(\beta_{\bar x}\), which is LT.
  5. (b). \(\mathrm{act}\) commutes with \(\mathrm{Fr}^*\), so it induces a map on twisted HH. On the spanning classes \([V_\lambda,\mathrm{can}]\) the formula is IIa.
  6. (c), cyclicity. With the relation \([M,g\circ\mathrm{Fr}^*f]=[N,f\circ g]\), \(f=\pi\), \(g=e\Phi\,\mathrm{Fr}^*\iota\), and \(e\Phi=\Phi\,\mathrm{Fr}^*e\): \(g\circ\mathrm{Fr}^*f=e\Phi\,\mathrm{Fr}^*e=e\Phi\) and \(f\circ g=\pi\iota\pi\Phi\,\mathrm{Fr}^*\iota=\pi\Phi\,\mathrm{Fr}^*\iota\). This is correct. Isotypic and generalized-eigen projectors are Frobenius-equivariant, and additivity along an \(A\)-stable flag gives \(d_\alpha\alpha[\mathrm{IC}_\nu,\mathrm{can}]\). Morita invariance on the \(\mathrm{Fr}^*\)-stable generating subcategory of sums of Satake objects is fine.
  7. (d), Barr–Beck. \(\mathrm{act}\) preserves compact objects, so \(\mathrm{act}^R\) is continuous and preserves all geometric realizations. \(\mathcal H=\operatorname{Thick}\{V_\lambda\}\) and the \(V_\lambda\) are dualizable, so \(\mathcal H\) is rigid and \(\mathrm{act}^R\) is \(\mathcal H\)-linear; hence \(\mathrm{act}^R\mathrm{act}\cong-\ast A\) as monads. Conservativity is equivalent to \(\{\mathrm{act}(F)\}\) generating \(\operatorname{Ind}\mathcal C_k\), i.e. to \(\operatorname{Thick}(\mathrm{act}\,\mathcal H)=\mathcal C_k\). HA Thm 4.7.3.5, checked locally, gives the equivalence with modules over the monad. The identification with Chen–Yi's (41) (\(A_X=\operatorname{RHom}(e_{L^+X},\mathrm{IC}_{\mathcal O(\check G)}\star e_{L^+X})\), checked locally) is plausible via \(A_X\simeq\bigoplus_\lambda\operatorname{Hom}(V_\lambda^\vee,A)\otimes V_\lambda^*\), and it is not used.
  8. Ponto–Shulman. Prop. 2.4 (cyclicity \(\operatorname{tr}(fg)=\operatorname{tr}(gf)\)) and Prop. 6.2 with Ex. 6.3 (\(\operatorname{tr}(f\otimes_RS)=\phi(\operatorname{tr}f)\)) say exactly what is claimed (checked in the local copy). They concern single endomorphisms.

Builds on

  • Theorem 1.3 · Theorem T: trace of Frobenius on \(\mathcal C_k\) is \(C_K\); the missing retract classes

Built on by

  • Summary 4.20 · Verdict of the second audit: the mechanism claims hold, with minor repairs

Sources